Let $f:[a,b]\rightarrow [a,b]$ be Riemann integrable with $a<c<b$. Show that $\int_a^cf$ and $\int_c^bf$ both exist. Use the definition of Riemann's criterion for integrability.
I don't really understand the question - surely if a function is Riemann integrable then its integral exists?
Let us focus on $[a, c]$.
Given any $\epsilon > 0,$ you wish to show that there exists a partition $\mathcal{P}$ of $[a, c]$ such that $$U(\mathcal{P}, f) - L(\mathcal{P}, f) < \epsilon.$$
Now, since $f$ is Riemann integrable on $[a, b]$, you know that that there exists a partition $\mathcal{Q}$ of $[a, b]$ such that $$U(\mathcal{Q}, f) - L(\mathcal{Q}, f) < \epsilon.$$
From this, we may first get a partition $\mathcal{Q}' = \mathcal{Q} \cup \{c\}$ of $[a, b]$. Note that this is a refinement and hence, $$U(\mathcal{Q}', f) - L(\mathcal{Q}', f) < \epsilon.$$
From the above, there is an obvious partition $\dot{\mathcal{Q}}$ of $[a, c]$ that you can get. Do you see what it is? Moreover, the above difference $(U - L)(\mathcal{Q}', f)$ is a certain sum of non-negative real numbers. When considering the corresponding difference for $\dot{\mathcal{Q}}$, we are just ignoring some of those terms. Thus, we get $$U(\dot{\mathcal{Q}}, f) - L(\dot{\mathcal{Q}}, f) < \epsilon,$$ as desired.